Subgroups — C₃×#10613647 #14439844

Order 1920.

712 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#66022	C₂	1	2	yes	maximal
2	C₃×C₅×#499	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#65353	C₂	1	2	yes	maximal, complemented
4	C₃×#65169	C₂	1	2	yes	maximal
5	C₃×#65163	C₂	1	2	yes	maximal, complemented
6	C₃×#65149	C₂	1	2	yes	maximal
7	C₃×#65144	C₂	1	2	yes	maximal, complemented
8	C₃×C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic
9	C₃×#64972	C₂	1	1	yes	maximal, characteristic, complemented
10	#10613647	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₃×C₅×#125	C₂²	1	2	yes	—
12	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
14	C₃×#4042	—	2	4	no	—
15	C₃×#4038	C₂²	1	4	yes	—
16	C₃×#4034	C₂²	1	4	yes	—
17	C₃×#4033	—	2	4	no	—
18	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
19	C₃×#4016	C₂²	1	2	yes	—
20	C₃×C₅×#107	C₂²	1	2	yes	—
21	C₃×C₅×#106	C₂²	1	2	yes	—
22	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
23	C₃×#3950	C₂²	1	2	yes	—
24	C₃×#3949	C₂²	1	2	yes	—
25	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
26	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
27	C₃×#3940	C₂²	1	4	yes	—
28	C₃×#3939	C₂²	1	4	yes	—
29	C₃×C₄×#168	C₂²	1	1	yes	characteristic
30	C₃×#3160	—	5	1	no	maximal, Hall
31	#66022	C₂×C₃	1	2	yes	—
32	C₅×#499	C₂×C₃	1	1	yes	characteristic, complemented
33	#65353	C₂×C₃	1	2	yes	complemented
34	#65169	C₂×C₃	1	2	yes	—
35	#65163	C₂×C₃	1	2	yes	complemented
36	#65149	C₂×C₃	1	2	yes	—
37	#65144	C₂×C₃	1	2	yes	complemented
38	C₄×C₈×D₅	C₂×C₃	1	1	yes	characteristic
39	#64972	C₂×C₃	1	1	yes	characteristic, complemented
40	C₂×C₃×C₄×D₅	D₄	1	2	yes	—
41	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
42	C₂×C₃×Dic₁₀	—	2	4	no	—
43	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
44	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
45	C₃×C₅×C₄⋊C₄	—	2	4	no	—
46	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
47	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
48	C₃×#685	—	2	4	no	—
49	C₃×C₄⋊Dic₅	C₂³	1	4	yes	—
50	C₃×#683	—	2	4	no	—
51	C₃×#683	—	2	4	no	—
52	C₃×C₄×Dic₅	—	2	4	no	—
53	C₃×C₄×Dic₅	D₄	1	2	yes	—
54	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
55	C₂×C₃×#168	C₂³	1	1	yes	characteristic
56	C₂×C₃×#168	C₂³	1	1	yes	characteristic
57	C₃×C₈×D₅	Q₈	1	4	yes	complemented
58	C₃×C₈×D₅	Q₈	1	4	yes	complemented
59	C₃×#533	—	5	2	no	—
60	C₃×#500	—	5	2	no	—
61	C₃×#499	—	5	2	no	—
62	C₃×#499	—	5	1	no	—
63	C₃×#499	D₅	1	1	yes	characteristic, complemented
64	C₃×#498	—	5	2	no	—
65	C₃×#427	—	5	2	no	—
66	C₃×#427	—	5	2	no	—
67	C₂×C₃×C₄×C₈	—	5	1	no	—
68	C₅×#125	C₂²×C₃	1	2	yes	—
69	C₂×C₈×D₅	C₂²×C₃	1	1	yes	characteristic
70	C₂×C₈×D₅	C₂²×C₃	1	1	yes	characteristic
71	#4042	—	2	4	no	—
72	#4038	C₂²×C₃	1	4	yes	—
73	#4034	C₂²×C₃	1	4	yes	—
74	#4033	—	2	4	no	—
75	C₄²×D₅	C₂²×C₃	1	1	yes	characteristic
76	#4016	C₂²×C₃	1	2	yes	—
77	C₅×#107	C₂²×C₃	1	2	yes	—
78	C₅×#106	C₂²×C₃	1	2	yes	—
79	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
80	#3950	C₂²×C₃	1	2	yes	—
81	#3949	C₂²×C₃	1	2	yes	—
82	C₈×Dic₅	C₂²×C₃	1	1	yes	characteristic
83	C₈×Dic₅	C₂²×C₃	1	1	yes	characteristic
84	#3940	C₂²×C₃	1	4	yes	—
85	#3939	C₂²×C₃	1	4	yes	—
86	C₄×#168	C₂²×C₃	1	1	yes	characteristic
87	#3160	—	5	1	no	Sylow
88	C₂²×C₃×D₅	C₂×D₄	1	1	yes	characteristic
89	C₂×C₃×C₄×C₅	—	2	4	no	—
90	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
91	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
92	C₂×C₃×Dic₅	—	2	4	no	—
93	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
94	C₂×C₃×Dic₅	C₂×D₄	1	2	yes	—
95	C₃×C₄×D₅	—	2	4	no	—
96	C₃×C₄×D₅	C₂×Q₈	1	2	yes	—
97	C₃×C₄×D₅	C₂×Q₈	1	2	yes	—
98	C₃×Dic₁₀	—	4	8	no	—
99	C₃×C₅×C₈	C₂×Q₈	1	2	yes	—
100	C₃×C₅×C₈	C₂×Q₈	1	2	yes	—
101	C₃×#168	C₂×Q₈	1	2	yes	—
102	C₃×#168	C₂×Q₈	1	2	yes	—
103	C₂²×C₃×C₈	—	5	1	no	—
104	C₂²×C₃×C₈	—	5	1	no	—
105	C₃×C₄⋊Q₈	—	5	4	no	—
106	C₃×#125	—	5	2	no	—
107	C₃×#125	C₂×D₅	1	2	yes	—
108	C₃×C₂²⋊Q₈	—	10	4	no	—
109	C₃×C₄×Q₈	—	10	4	no	—
110	C₃×#117	—	5	4	no	—
111	C₂×C₃×C₄²	—	5	1	no	—
112	C₃×#107	—	5	2	no	—
113	C₃×#107	—	5	4	no	—
114	C₃×#107	C₂×D₅	1	2	yes	—
115	C₃×#106	—	5	2	no	—
116	C₃×#106	—	5	4	no	—
117	C₃×#106	C₂×D₅	1	2	yes	—
118	C₃×C₄×C₈	—	5	1	no	—
119	C₃×C₄×C₈	—	5	1	no	—
120	C₃×C₄×C₈	—	5	1	no	—
121	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
122	C₂×C₄×D₅	C₃×D₄	1	2	yes	—
123	C₂×C₄×D₅	C₂³×C₃	1	1	yes	characteristic
124	C₂×Dic₁₀	—	2	4	no	—
125	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
126	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
127	C₅×C₄⋊C₄	—	2	4	no	—
128	C₅×C₄⋊C₄	C₂³×C₃	1	4	yes	—
129	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
130	#685	—	2	4	no	—
131	C₄⋊Dic₅	C₂³×C₃	1	4	yes	—
132	#683	—	2	4	no	—
133	#683	—	2	4	no	—
134	C₄×Dic₅	—	2	4	no	—
135	C₄×Dic₅	C₃×D₄	1	2	yes	—
136	C₄×Dic₅	C₂³×C₃	1	1	yes	characteristic
137	C₂×#168	C₂³×C₃	1	1	yes	characteristic
138	C₂×#168	C₂³×C₃	1	1	yes	characteristic
139	C₈×D₅	C₃×Q₈	1	4	yes	complemented
140	C₈×D₅	C₃×Q₈	1	4	yes	complemented
141	#533	—	5	2	no	—
142	#500	—	5	2	no	—
143	#499	—	5	2	no	—
144	#499	—	5	1	no	—
145	#499	C₃×D₅	1	1	yes	characteristic, complemented
146	#498	—	5	2	no	—
147	#427	—	5	2	no	—
148	#427	—	5	2	no	—
149	C₂×C₄×C₈	—	5	1	no	—
150	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₃×D₅	—	2	2	no	—
152	C₂×C₃×D₅	C₄⋊Q₈	1	2	yes	—
153	C₃×C₄×C₅	—	2	2	no	—
154	C₃×C₄×C₅	—	4	4	no	—
155	C₃×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
156	C₃×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
157	C₃×Dic₅	—	4	4	no	—
158	C₃×Dic₅	C₄⋊Q₈	1	2	yes	—
159	C₃×Dic₅	#135	1	4	yes	—
160	C₂×C₃×Q₈	—	10	4	no	—
161	C₂²×C₃×C₄	—	5	1	no	—
162	C₂²×C₃×C₄	—	5	2	no	—
163	C₂×C₃×C₈	—	5	1	no	—
164	C₂×C₃×C₈	—	5	1	no	—
165	C₂×C₃×C₈	—	5	4	no	—
166	C₂×C₃×C₈	—	5	4	no	—
167	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
168	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
169	C₃×C₄⋊C₄	—	2	4	no	—
170	C₃×C₄⋊C₄	—	5	4	no	—
171	C₃×C₄⋊C₄	—	10	4	no	—
172	C₃×C₄⋊C₄	—	10	4	no	—
173	C₃×C₄⋊C₄	C₂²×D₅	1	4	yes	—
174	C₃×C₂²⋊C₄	—	10	4	no	—
175	C₃×C₄²	—	5	2	no	—
176	C₃×C₄²	—	5	1	no	—
177	C₃×C₄²	—	10	4	no	—
178	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
179	C₂²×D₅	C₂×C₃×D₄	1	1	yes	characteristic
180	C₂×C₄×C₅	—	2	4	no	—
181	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
182	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
183	C₂×Dic₅	—	2	4	no	—
184	C₂×Dic₅	C₂×C₃×D₄	1	1	yes	characteristic
185	C₂×Dic₅	C₂×C₃×D₄	1	2	yes	—
186	C₄×D₅	—	2	4	no	—
187	C₄×D₅	C₂×C₃×Q₈	1	2	yes	—
188	C₄×D₅	C₂×C₃×Q₈	1	2	yes	—
189	Dic₁₀	—	4	8	no	—
190	C₅×C₈	C₂×C₃×Q₈	1	2	yes	—
191	C₅×C₈	C₂×C₃×Q₈	1	2	yes	—
192	#168	C₂×C₃×Q₈	1	2	yes	—
193	#168	C₂×C₃×Q₈	1	2	yes	—
194	C₂²×C₈	—	5	1	no	—
195	C₂²×C₈	—	5	1	no	—
196	C₄⋊Q₈	—	5	4	no	—
197	#125	—	5	2	no	—
198	#125	C₂×C₃×D₅	1	2	yes	—
199	C₂²⋊Q₈	—	10	4	no	—
200	C₄×Q₈	—	10	4	no	—
201	#117	—	5	4	no	—
202	C₂×C₄²	—	5	1	no	—
203	#107	—	5	2	no	—
204	#107	—	5	4	no	—
205	#107	C₂×C₃×D₅	1	2	yes	—
206	#106	—	5	2	no	—
207	#106	—	5	4	no	—
208	#106	C₂×C₃×D₅	1	2	yes	—
209	C₄×C₈	—	5	1	no	—
210	C₄×C₈	—	5	1	no	—
211	C₄×C₈	—	5	1	no	—
212	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic
213	C₂×C₃×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
214	C₂×C₃×C₅	C₂×#135	1	2	yes	—
215	C₃×D₅	—	2	2	no	—
216	C₂³×C₃	—	5	1	no	—
217	C₃×Q₈	—	20	8	no	—
218	C₂×C₃×C₄	—	2	4	no	—
219	C₂×C₃×C₄	—	5	1	no	—
220	C₂×C₃×C₄	—	5	2	no	—
221	C₂×C₃×C₄	—	5	2	no	—
222	C₂×C₃×C₄	—	5	2	no	—
223	C₂×C₃×C₄	—	10	4	no	—
224	C₂×C₃×C₄	—	10	4	no	—
225	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
226	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
227	C₃×C₈	—	5	2	no	—
228	C₃×C₈	—	5	2	no	—
229	C₃×C₈	Q₈×D₅	1	2	yes	—
230	C₃×C₈	Q₈×D₅	1	2	yes	—
231	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
232	C₂×D₅	—	2	2	no	—
233	C₂×D₅	C₃×C₄⋊Q₈	1	2	yes	—
234	C₄×C₅	—	2	2	no	—
235	C₄×C₅	—	4	4	no	—
236	C₄×C₅	C₂²×C₃×Q₈	1	1	yes	characteristic
237	C₄×C₅	C₂²×C₃×Q₈	1	1	yes	characteristic
238	Dic₅	—	4	4	no	—
239	Dic₅	C₃×C₄⋊Q₈	1	2	yes	—
240	Dic₅	C₃×#135	1	4	yes	—
241	C₂×Q₈	—	10	4	no	—
242	C₂²×C₄	—	5	1	no	—
243	C₂²×C₄	—	5	2	no	—
244	C₂×C₈	—	5	1	no	—
245	C₂×C₈	—	5	1	no	—
246	C₂×C₈	—	5	4	no	—
247	C₂×C₈	—	5	4	no	—
248	C₂×C₈	C₂²×C₃×D₅	1	1	yes	characteristic
249	C₂×C₈	C₂²×C₃×D₅	1	1	yes	characteristic
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	5	4	no	—
252	C₄⋊C₄	—	10	4	no	—
253	C₄⋊C₄	—	10	4	no	—
254	C₄⋊C₄	C₂²×C₃×D₅	1	4	yes	—
255	C₂²⋊C₄	—	10	4	no	—
256	C₄²	—	5	2	no	—
257	C₄²	—	5	1	no	—
258	C₄²	—	10	4	no	—
259	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
260	C₃×C₅	#3160	1	1	yes	characteristic, complemented, Hall
261	C₂²×C₃	—	5	2	no	—
262	C₂²×C₃	—	10	2	no	—
263	C₂²×C₃	C₂×D₄×D₅	1	1	yes	characteristic
264	C₃×C₄	—	2	2	no	—
265	C₃×C₄	—	4	4	no	—
266	C₃×C₄	—	5	2	no	—
267	C₃×C₄	—	5	4	no	—
268	C₃×C₄	—	20	4	no	—
269	C₃×C₄	C₂×Q₈×D₅	1	1	yes	characteristic
270	C₃×C₄	C₂×Q₈×D₅	1	1	yes	characteristic
271	C₂×C₅	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
272	C₂×C₅	C₂×C₃×#135	1	2	yes	—
273	D₅	—	2	2	no	—
274	C₂³	—	5	1	no	—
275	Q₈	—	20	8	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	5	1	no	—
278	C₂×C₄	—	5	2	no	—
279	C₂×C₄	—	5	2	no	—
280	C₂×C₄	—	5	2	no	—
281	C₂×C₄	—	10	4	no	—
282	C₂×C₄	—	10	4	no	—
283	C₂×C₄	C₃×D₄×D₅	1	2	yes	—
284	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
285	C₈	—	5	2	no	—
286	C₈	—	5	2	no	—
287	C₈	C₃×Q₈×D₅	1	2	yes	—
288	C₈	C₃×Q₈×D₅	1	2	yes	—
289	C₂×C₃	—	10	2	no	—
290	C₂×C₃	D₅×C₄⋊Q₈	1	1	yes	characteristic
291	C₂×C₃	D₅×#135	1	2	yes	—
292	C₅	C₃×#3160	1	1	yes	characteristic, complemented, Sylow
293	C₂²	—	5	2	no	—
294	C₂²	—	10	2	no	—
295	C₂²	C₂×C₃×D₄×D₅	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	4	4	no	—
298	C₄	—	5	4	no	—
299	C₄	—	5	2	no	—
300	C₄	—	20	4	no	—
301	C₄	C₂×C₃×Q₈×D₅	1	1	yes	characteristic
302	C₄	C₂×C₃×Q₈×D₅	1	1	yes	characteristic
303	C₃	#10613647	1	1	yes	characteristic, direct product factor, Sylow
304	C₂	—	10	2	no	—
305	C₂	C₃×D₅×C₄⋊Q₈	1	1	yes	characteristic
306	C₂	C₃×D₅×#135	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂×D₄×D₅
commutator, cosocle	C₂×C₄×C₅	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₅
Fitting	C₃×C₅×#499	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂²×C₃×D₅	C₂×D₄
hypercenter	C₃×#499	D₅
hypocenter	C₅	C₃×#3160
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#499

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₃×C₅×#499	C₂	semidirect
3. C₃×#65353	C₂	semidirect
5. C₃×#65163	C₂	semidirect
7. C₃×#65144	C₂	semidirect
9. C₃×#64972	C₂	semidirect
10. #10613647	C₃	direct (N × H)
32. C₅×#499	C₂×C₃	semidirect
33. #65353	C₂×C₃	semidirect
35. #65163	C₂×C₃	semidirect
37. #65144	C₂×C₃	semidirect
39. #64972	C₂×C₃	semidirect
57. C₃×C₈×D₅	Q₈	semidirect
58. C₃×C₈×D₅	Q₈	semidirect
63. C₃×#499	D₅	semidirect
139. C₈×D₅	C₃×Q₈	semidirect
140. C₈×D₅	C₃×Q₈	semidirect
145. #499	C₃×D₅	semidirect
260. C₃×C₅	#3160	semidirect
292. C₅	C₃×#3160	semidirect
303. C₃	#10613647	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×#66022	2
C₃×C₅×#499	2
C₃×#65353	2
C₃×#65169	2
C₃×#65163	2
C₃×#65149	2
C₃×#65144	2
C₃×C₄×C₈×D₅	2
C₃×#64972	2
#10613647	3
C₃×#3160	5

Subgroups — C3×#10613647 #14439844

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10613647 #14439844